A city has $4$ horizontal and $n\geq3$ vertical boulevards which intersect at $4n$ crossroads. The crossroads divide every horizontal boulevard into $n-1$ streets and every vertical boulevard into $3$ streets. The mayor of the city decides to close the minimum possible number of crossroads so that the city doesn't have a closed path(this means that starting from any street and going only through open crossroads without turning back you can't return to the same street). $a)$Prove that exactly $n$ crossroads are closed. $b)$Prove that if from any street you can go to any other street and none of the $4$ corner crossroads are closed then exactly $3$ crossroads on the border are closed(A crossroad is on the border if it lies either on the first or fourth horizontal boulevard, or on the first or the n-th vertical boulevard).

A finite set $V \in Z^2$ of vectors with integer coordinates is chosen on the plane. Each of them is painted one of the $n$ colors. The color is [i]suitable[/i] for the vector if this vector may be presented as' a linear combination (with integer coefficients) of the vectors from $V$ of this color. It is known,that for any vector from $Z^2$ there exist a suitable color. Find all $n$ such that there must exist a color which is suitable for any vector from $Z^2$ . (V. Lebed)

Given a positive integer $n$, show that for no set of integers modulo $n$, whose size exceeds $1+\sqrt{n+4}$, is it possible that the pairwise sums of unordered pairs be all distinct.

Prove that, given a positive integrer $n$, there exists a positive integrer $k_n$ with the following property: Given any $k_n$ points in the space, $4$ by $4$ non-coplanar, and associated integrer numbers between $1$ and $n$ to each sharp edge that meets $2$ of this points, there's necessairly a triangle determined by $3$ of them, whose sharp edges have associated the same number.

For all integers $x,y$, a non-negative integer $f(x,y)$ is written on the point $(x,y)$ on the coordinate plane. Initially, $f(0,0) = 4$ and the value written on all remaining points is $0$. For integers $n, m$ that satisfies $f(n,m) \ge 2$, define '[color=#9a00ff]Seehang[/color]' as the act of reducing $f(n,m)$ by $1$, selecting 3 of $f(n,m+1), f(n,m-1), f(n+1,m), f(n-1,m)$ and increasing them by 1. Prove that after a finite number of '[color=#0f0][color=#9a00ff]Seehang[/color][/color]'s, it cannot be $f(n,m)\le 1$ for all integers $n,m$.

Bob plotted $2003$ green points on the plane, so all triangles with three green vertices have area less than $1$. Prove that the $2003$ green points are contained in a triangle $T$ of area less than $4$.

Place 2005 points on the circumference of a circle. Two points $P,Q$ are said to form a pair of neighbours if the chord $PQ$ subtends an angle of at most 10 degrees at the centre. Find the smallest number of pairs of neighbours.

Morteza and Amir Reza play the following game. First each of them independently roll a dice $100$ times in a row to construct a $100$-digit number with digits $1,2,3,4,5,6$ then they simultaneously shout a number from $1$ to $100$ and write down the corresponding digit to the number other person shouted in their $100$ digit number. If both of the players write down $6$ they both win otherwise they both loose. Do they have a strategy with wining chance more than $\frac{1}{36}$? [i]Proposed by Morteza Saghafian[/i]

Players $A$ and $B$ play the following game on the coordinate plane. Player $A$ hides a nut at one of the points with integer coordinates, and player $B$ tries to find this hidden nut. In one move $B$ can choose three different points with integer coordinates, then $A$ tells whether these three points together with the nut's point lie on the same circle or not. Can $B$ be guaranteed to find the nut in a finite number of moves?

Let $N$ be the number of ways to colour each cell in a $2 \times 50$ rectangle either red or blue such that each $2 \times 2$ block contains at least one blue cell. Show that $N$ is a multiple of $3^{25}$, but not a multiple of $3^{26}$

The $2013$ numbers $$\frac{1}{1\times 2}, \frac{1}{2\times 3},\frac{1}{3\times 4},...,\frac{1}{2013 \times 2014}$$ are arranged randomly on a circle. (a) Prove that there exist ten consecutive numbers on the circle whose sum is less than $\frac{1}{4000}$ . (b) Prove that there exist ten consecutive numbers on the circle whose sum is less than $\frac{1}{10000}$ .

The following expression $x^{30} + *x^{29} +...+ *x+8 = 0$ is written on a blackboard. Two players $A$ and $B$ play the following game. $A$ starts the game. He replaces all the asterisks by the natural numbers from $1$ to $30$ (using each of them exactly once). Then player $B$ replace some of" $+$ "by ” $-$ "(by his own choice). The goal of $A$ is to get the equation having a real root greater than $10$, while the goal of $B$ is to get the equation having a real root less that or equal to $10$. If both of the players achieve their goals or nobody of them achieves his goal, then the result of the game is a draw. Otherwise, the player achieving his goal is a winner. Who of the players wins if both of them play to win? (I.Bliznets)

In a lake, there are 2024 leaves arranged in a row. Two frogs are positioned, one on the first leaf and the other on the second leaf. Every minute, both frogs jump simultaneously. Each time a frog jumps, it decides whether to jump to the next leaf or to the leaf that is three positions ahead. Is it possible for each leaf to be visited exactly once by exactly one of the frogs?

The localities $P_1, P_2, \dots, P_{1983}$ are served by ten international airlines $A_1,A_2, \dots , A_{10}$. It is noticed that there is direct service (without stops) between any two of these localities and that all airline schedules offer round-trip flights. Prove that at least one of the airlines can offer a round trip with an odd number of landings.

Let $n \geq2$ be an integer number and $D_n$ the set of all the points $(x,y)$ in the plane such that its coordinates are integer numbers with: $-n \le x \le n$ and $-n \le y \le n$. (a) There are three possible colors in which the points of $D_n$ are painted with (each point has a unique color). Show that with any distribution of the colors, there are always two points of $D_n$ with the same color such that the line that contains them does not go through any other point of $D_n$. (b) Find a way to paint the points of $D_n$ with 4 colors such that if a line contains exactly two points of $D_n$, then, this points have different colors.

Given a number $n\in\mathbb{Z}^+$ and let $S$ denotes the set $\{0,1,2,...,2n+1\}$. Consider the function $f:\mathbb{Z}\times S\to [0,1]$ satisfying two following conditions simultaneously: i) $f(x,0)=f(x,2n+1)=0\forall x\in\mathbb{Z}$; ii) $f(x-1,y)+f(x+1,y)+f(x,y-1)+f(x,y+1)=1$ for all $x\in\mathbb{Z}$ and $y\in\{1,2,3,...,2n\}$. Let $F$ be the set of such functions. For each $f\in F$, let $v(f)$ be the set of values of $f$. a) Proof that $|F|=\infty$. b) Proof that for each $f\in F$ then $|v(f)|<\infty$. c) Find the maximum value of $|v(f)|$ for $f\in F$.

Ten children have several bags of candies. The children begin to divide these candies among them. They take turns picking their shares of candies from each bag, and leave just after that. The size of the share is determined as follows: the current number of candies in the bag is divided by the number of remaining children (including the one taking the turn). If the remainder is nonzero than the quotient is rounded to the lesser integer. Is it possible that all the children receive different numbers of candies if the total number of bags is: a) 8 ; 6) 99 ? Alexey Glebov

a) There are $2n$ rays marked in a plane, with $n$ being a natural number. Given that no two marked rays have the same direction and no two marked rays have a common initial point, prove that there exists a line that passes through none of the initial points of the marked rays and intersects with exactly $n$ marked rays. (b) Would the claim still hold if the assumption that no two marked rays have a common initial point was dropped?

Show that there is an absolute constant $c < 1$ with the following property: whenever $\mathcal P$ is a polygon with area $1$ in the plane, one can translate it by a distance of $\frac{1}{100}$ in some direction to obtain a polygon $\mathcal Q$, for which the intersection of the interiors of $\mathcal P$ and $\mathcal Q$ has total area at most $c$. [i]Linus Hamilton[/i]

Show that there is a set of $2002$ consecutive positive integers containing exactly $150$ primes. (You may use the fact that there are $168$ primes less than $1000$)

There are $100$ piles of $400$ stones each. At every move, Pete chooses two piles, removes one stone from each of them, and is awarded the number of points, equal to the non- negative difference between the numbers of stones in two new piles. Pete has to remove all stones. What is the greatest total score Pete can get, if his initial score is $0$? (Maxim Didin)

$300$ parliament members are divided into $3$ chambers, each chamber consists of $100$ members. For every $2$ members, they either know each other or are strangers to each other.Show that no matter how they are divided into these $3$ chambers, it is always possible to choose $2$ members, each from different chamber such that there exist $17$ members from the third chamber so that all of them knows these two members, or all of them are strangers to these two members.

There are distinct points $O, A, B, K_1, . . . , K_n, L_1, . . . , L_n$ on a plane such that no three points are collinear. The open line segments $K_1L_1, . . . , K_nL_n$ are coloured red, other points on the plane are left uncoloured. An allowed path from point $O$ to point $X$ is a polygonal chain with first and last vertices at points $O$ and $X$, containing no red points. For example, for $n = 1$, and $K_1 = (-1, 0)$, $L_1 = (1, 0)$, $O = (0,-1)$, and $X = (0,1)$, $OK_1X$ and $OL_1X$ are examples of allowed paths from $O$ to $X$, there are no shorter allowed paths. Find the least positive integer n such that it is possible that the first vertex that is not $O$ on any shortest possible allowed path from $O$ to $A$ is closer to $B$ than to $A$, and the first vertex that is not $O$ on any shortest possible allowed path from $O$ to $B$ is closer to $A$ than to $B$.

For each set $C$ of points in space, we designate by $P_C$ the set of planes containing at least three points of $C$. $\bullet$ Prove that there exists $C$ such that $\phi (P_C) = 1997$, where $\phi$ corresponds to the cardinality. $\bullet$ Determine the least number of points that $C$ must have so that the previous property can be fulfilled.

There are $ n$ voters and $ m$ candidates. Every voter makes a certain arrangement list of all candidates (there is one person in every place $ 1,2,...m$) and votes for the first $ k$ people in his/her list. The candidates with most votes are selected and say them winners. A poll profile is all of this $ n$ lists. If $ a$ is a candidate, $ R$ and $ R'$ are two poll profiles. $ R'$ is $ a\minus{}good$ for $ R$ if and only if for every voter; the people which in a worse position than $ a$ in $ R$ is also in a worse position than $ a$ in $ R'$. We say positive integer $ k$ is monotone if and only if for every $ R$ poll profile and every winner $ a$ for $ R$ poll profile is also a winner for all $ a\minus{}good$ $ R'$ poll profiles. Prove that $ k$ is monotone if and only if $ k>\frac{m(n\minus{}1)}{n}$.